A car of mass 1396 kg is parked on a very slippery hillside.To keep it from sliding down the hill (è = 15°), the owner attaches a cable. What is the tension in the cable?Determine the components of all the forces on the car along the coordinate axes.
To find the tension in the cable, we need to analyze the forces acting on the car. Given that the car is parked on a very slippery hillside, the main forces acting on the car are gravity and the tension in the cable.
Here's how we can determine the tension in the cable:
1. Break down the weight of the car into its components along the coordinate axes. The weight of the car is given by the equation:
Weight = mass x acceleration due to gravity
Weight = 1396 kg x 9.8 m/s² (acceleration due to gravity ≈ 9.8 m/s²)
Weight = 13652.8 N
The weight can be broken down into two components:
- The component parallel to the hillside (mgsinθ): This component acts in the downhill direction and tends to pull the car down the hill. Here, θ refers to the angle of the hillside (15°), and m refers to the mass (1396 kg).
- The component perpendicular to the hillside (mgcosθ): This component acts perpendicular to the hillside and doesn't contribute to the sliding motion. It is balanced by the normal force exerted by the hillside on the car.
2. Determine the tension in the cable. Since the car is not sliding down the hill, the tension in the cable must be equal in magnitude but opposite in direction to the downhill component of the weight. Therefore, the tension can be given as:
Tension in the cable (T) = mgsinθ
T = 1396 kg x 9.8 m/s² x sin(15°)
T = 3308.52 N
So, the tension in the cable is approximately 3308.52 N.
3. Determine the components of all the forces on the car along the coordinate axes:
- The downhill force (T) acts in the negative y-direction.
- The perpendicular force (mgcosθ) acts in the negative x-direction.
These are the main forces acting on the car along the coordinate axes.